**1**

vote

**1**answer

103 views

### Rigid effective divisors

Let $D\subset X$ be an effective smooth divisor in a smooth projective variety $X$. Assume that $h^0(X,D)=1$. In particular $D$ spans an extremal ray of the effective cone of $X$.
Now, let ...

**2**

votes

**0**answers

82 views

### How the existence of holomorphic sections depends on the choice of complex structure

In this Mathoverflow question it is asked how many invariant complex structures exist on the full flag manifold of $SU(m)$. In this question it is asked when a line bundle over a flag manifold has ...

**2**

votes

**1**answer

116 views

### Pencils in very ample linear systems without curve in its base locus

If $L$ is a very ample line bundle over a smooth complex projective surface $X$ and $s_0, \dots, s_n$ is a basis of the global sections of $L$, is there some choice of $i,j$ such that the pencil ...

**5**

votes

**1**answer

166 views

### Pushforward of line bundle under “toric isogeny”

Let $(X,T)$ be a smooth complex toric variety of dimension $d$ with torus $T$ and toric boundary $D=X\setminus T$. Let $\phi : X\to X$ be a finite endomorphism of $X$ such that the restriction
...

**1**

vote

**0**answers

100 views

### Line bundles with vanishing cohomology on Calabi-Yau manifold

Suppose we have some line bundle $L(D)$ on Calabi-Yau threefold. Let's call this line bundle "rigid" if $H^0(X,L(D)) \simeq \mathbb{C}$ and $H^i(X,L(D))=0$ for $i=1,2,3$.
Is anything known about such ...

**3**

votes

**1**answer

213 views

### Picard groups, ample cones, and proper birational maps

Let $f:Y\to X$ be a proper birational map of normal varieties over an algebraically closed field which is an isomorphism over the regular locus.
Q1: Is it the case that the pullback ...

**2**

votes

**0**answers

126 views

### Universal property of limits of invertible sheaves

Let $R$ be a discrete valuation ring, $m$ the maximal ideal and $f:X \to \mathrm{Spec}(R)$ be a flat, proper morphism of relative dimension $1$. Assume further that $X$ is regular. For any $n>0$, ...

**1**

vote

**0**answers

124 views

### Are two line bundles with the same ramification type necessarily isomorphic?

I have no motivation for the following problem, I am just curious if it is true or not. Here it is:
If $l_1$ and $l_2$ are two complete $g^r_d$'s on a smooth curve $C$ such that the vanishing ...

**2**

votes

**2**answers

235 views

### Could we extend any line bundle on the smooth part of a singular curve to a line bundle on the whole curve?

Let $X$ be a singular curve over an algebraic closed field $k$ with characteristic zero. Let $Z$ be the closed subset of singular points on $X$ and $U=X-Z$ be the smooth part, which is an open subset ...

**6**

votes

**2**answers

324 views

### Will (general points + small number of arbitrary points) impose independent condtions on plane curves?

It is well known that imposing vanishing at general points of $\mathbb P^2$ gives independent conditions on curves of degree $d$. Also, it is known that a small number ($\le d+1$) points always impose ...

**5**

votes

**0**answers

333 views

### Correspondence between line bundles and $U(1)$-bundles: a possible mistake from physicists? [closed]

I am reading a paper written by physicists and they say the following:
Let $(L,h,\nabla)$ be an holomorphic Line bundle equipped with a hermitian metric $h$ and Chern connection $\nabla$. If ...

**2**

votes

**1**answer

181 views

### Computing Euler Characteristics of Line Bundles on the Hilbert Scheme of n points

Let $S^{[n]}$ be the Hilbert scheme of $n$ points on a smooth projective surface (actually, right now I am particularly interested in del Pezzo surfaces). Let $B$ be the exceptional divisor of the ...

**0**

votes

**1**answer

394 views

### Is there a toplogically trivial line bundle over a compact Riemann surfaces that isn't holomorphically trivial? [closed]

Is a complex line bundle over a compact Riemann surface topologically trivial iff it is holomorphically trivial? If so, how does one demonstrate that, and if not, what is a counterexample?

**2**

votes

**3**answers

592 views

### Movable Divisors

Let $X$ be a projective variety. Does anyone know an example of a movable reducible divisor $D\in Mov(X)$ such that any element in the linear system $|D|$ of $D$ is reducible?

**6**

votes

**4**answers

849 views

### Cone over the Veronese surface

Let $V\subset\mathbb{P}^5$ be the Veronese surface and let $X\subset\mathbb{P}^6$ be the cone over it. Since $X$ is $\mathbb{Q}$-factorial there are two integers $a,b$ such that $aK_X = ...

**2**

votes

**1**answer

187 views

### How to see that this pairing of line bundles is multiplicative?

Given a projective flat morphism $p: X \rightarrow Y$ of integral noetherian schemes of relative dimension one.
For a coherent sheaf $F$ on $Y$ we can define a line bundle $det(F)$ on $Y$ and for a ...

**0**

votes

**0**answers

113 views

### When can one find holomorphic sections vanishing at a point to a certain order?

Let $X$ be a compact complex manifold (say of dimension $2$) and $L \rightarrow X $ a holomorphic line bundle. Consider the following statements:
Statement $A_0$: Given any point $p\in X$, there ...

**2**

votes

**0**answers

109 views

### Computing Euler Charactistics of Line bundles on Hilbert Schemes of points on Surfaces

Let $S^{[2]}$ be the Hilbert scheme of two points on a smooth projective surface (actually, right now I am particularly interested in del Pezzo surfaces). Let $B$ be the exceptional divisor of the ...

**2**

votes

**2**answers

179 views

### Holomorphic Line Bundles over a Homogeneous Space

Let $M=G/H$ be (compact) homogeneous complex manifold, and let $L$ be a line bundle over $M$. Can one always equip $L$ with a holomorphic structure? Can there be more then one such holomorphic ...

**1**

vote

**1**answer

126 views

### Perfectness of the Jacobian of a curve

Let $C$ be a smooth projective curve over a field $K$ of characteristic $0$ (but not necessarily algebraically closed). Let $\mathcal{L}$ be a line bundle on $C$ of degree $0$. Fix an integer ...

**15**

votes

**1**answer

552 views

### Maps to projective space == line bundles; what do maps to weighted projective space correspond to?

A map from an algebraic variety $X$ to a projective space is the same thing as a globally generated line bundle on $X$. What geometric object on $X$ corresponds to a map to a weighted projective ...

**1**

vote

**1**answer

133 views

### Big divisors and small transformations

Let $X$ be a smooth projective variety such that $-K_X$ is ample. Let $f:X\dashrightarrow Y$ be a small $\mathbb{Q}$-factorial transformation. I would like to know if is true or not that:
$-K_Y$ is ...

**1**

vote

**2**answers

758 views

### Big and Nef divisors

In Example 2.2.19 of
Lazarsfeld, Positivity in Algebraic Geometry I,
I found the following statement:
Let $D$ be a divisor on an irreducible projective variety $X$. Then $D$ is nef and big if and ...

**0**

votes

**1**answer

82 views

### Two questions about line bundles over Kuranishi families

i'm studying the article "Variétés Kahleriennes dont la première classe de chern est nulle" by Arnaud Beauville and i have a couple of questions i would like to ask you, hoping they are not too ...

**2**

votes

**1**answer

112 views

### When is the Clifford index of a curve computed by pencils?

Under which circumstances is the Clifford index of a curve computed by pencils?

**-1**

votes

**2**answers

194 views

### Restriction of a line bundle to a two-cycle

I am reading a paper on Chiral Differential Operators
http://arxiv.org/pdf/hep-th/0604179v3.pdf
and it says on page 23 that a line bundle over a manifold C can be characterized completely by its ...

**0**

votes

**1**answer

202 views

### Birkhoff decomposition vanishing of the Chern numbers

Birkhoff decomposition vanishing of the Chern numbers of the holomorphic line bundles of the Birkhoff-Grothendieck decomposition, is some statement I read off in One of Connes papers. Without going ...

**1**

vote

**1**answer

100 views

### Unicity of a vector field on $S^1$-bundle

Let M be a complex smooth manifold,and let $\zeta $ be a vector filed on $M$, why always there exists a unique vector field $\hat{\zeta }$ on $L^{\times}$ which project down to $\zeta $ and $\alpha( ...

**0**

votes

**4**answers

326 views

### An isomorphism on space of smooth sections

Let $M$ be a smooth complex manifold and $L$ be a complex line bundle over $M$. Let $\Gamma(M,L)$ be the space of smooth sections. Why $\Gamma(M,L)$ is it isomorphic to
$$A=\{f:L^{\times}\to ...

**0**

votes

**2**answers

336 views

### Isomorphism of connections on a complex line bundle

Reading an article I faced with the following theorem, please give me a reference to a proof of the fact which is stated without any reference in the article. Is it a well-known fact?
Theorem. Let $E ...

**0**

votes

**2**answers

612 views

### Uniqueness on square root of complex Line Bundle

Let $L$ be a line bundle over a compex manifold $X$, a square-root of $L$ is a line bundle $M$ such that $M^{\otimes2}=L$. My question is when the square-root of Line Bundle is unique?

**1**

vote

**0**answers

149 views

### Hopf lemma for line bundles on curves in algebraic geometry

In the paper http://arxiv.org/pdf/math/0110256v1.pdf Claire Voisin proves that all linear subspaces which lie inside of a (not too big) secant variety of a smooth projective curve must lie inside one ...

**0**

votes

**0**answers

127 views

### canonical bundle of the relative spectrum

maybe it is a very trivial quetion but:
suppose we have a smooth projective variety $X$ over $k$ and $\mathcal{A}$ an $\mathcal{O}_X$ algebra. We have the relative spectrum ...

**0**

votes

**1**answer

226 views

### Could we construct the Jacobian variety of a smooth curve $C$ with genus $>2$ from its derived category $D(C)$?

Let's consider a smooth curve $C$ over $\mathbb{C}$. We know that the Jacobian variety $Jac(C)$ of $C$ is the moduli space of the degree $0$ line bundles on $C$. $Jac(C)$ is an abelian variety of ...

**3**

votes

**0**answers

332 views

### Ample Line Bundles on Algebraic Spaces

The sources known to me (Knutson's Algebraic Spaces and Pascual-Gainza's Ampleness criteria for algebraic spaces) define a line bundle $L$ on an algebraic space $X$ (over a base scheme $S$) to be ...

**1**

vote

**1**answer

242 views

### extension of torsion line bundles

Assume that you are given a smooth projective algebraic variety $X$ and a divisor $D$ with normal crossings on it. Put $U=X-D$. Assume now that $L$ is a torsion line bundle(=invertible sheaf) on $U$, ...

**8**

votes

**1**answer

483 views

### Vanishing theorems in positive characteristic

In the paper
Deligne, Pierre; Illusie, Luc (1987), "Relèvements modulo $p^{2}$ et décomposition du complexe de De Rham", Inventiones Mathematicae 89 (2): 247–270, doi:10.1007/BF01389078
I found the ...

**16**

votes

**3**answers

2k views

### What is Kirillov's method good for?

I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory important for ...

**0**

votes

**1**answer

151 views

### Trivial Line Bundle-Riemann surfaces

What are the Hermitian metrics in a trivial line bundle on a Riemann surface X?
I read that a Hermitian metric in the trivial line bundle is equivalent to a $\mathcal{C}^{\infty}$ weight function ...

**1**

vote

**0**answers

206 views

### canonical model of a reducible curve

Let $C$ be a stable reducible curve. Is there a natural way to define it's canonical model (I guess via the dualizing sheaf)? And does somehow the dualizing sheaf restrict to the (probably twisted) ...

**3**

votes

**2**answers

358 views

### Line bundles on K3 surfaces

Let $L$ be a line bundle on an (algebraic) K3 surface over a field $k$. The Riemann-Roch theorem specializes to
$$
\chi(X, L)=\frac{1}{2}(L\cdot L)+2
$$
which can be rewritten as
$$
h^0(X, ...

**9**

votes

**2**answers

880 views

### How many flat connections has a line bundle in algebraic geometry?

Suppose $X$ is a projective variety over $\mathbb C$. I am happy to entertain more or different adjectives — I'm not looking for the most general statement, but rather to understand when and how ...

**2**

votes

**1**answer

460 views

### On morphisms to projective space arising from a linear system

Context: This question arose as I was reading the proof of Application 6.1 in Mumford's Abelian Varieties. However, I have extracted all of the relevant information below so this question should ...

**1**

vote

**1**answer

241 views

### Extending line bundles

Suppose you have a one parameter family of algebraic varieties over unit disk, such that the central fiber is singular and is a union (normal-crossing) of two varieties and the rest are smooth.
Is it ...

**2**

votes

**1**answer

191 views

### divisors and powers of line bundles

Can anyone help me with the following question? Let $X$ be a smooth, projective algebraic variety over a field $k$ of characteristic zero. Let $D$ be an effective divisor on $X$ and $m \geq 2$ an ...

**1**

vote

**0**answers

207 views

### How does the line bundles look like on a proper model (or Néron model) of an abelian variety?

How does the line bundles look like on a proper model (or Néron model) of an abelian variety?
Who knows references about this?
In particular, let us work over a trait $S=\mathrm{Spec} R$, where $R$ ...

**7**

votes

**1**answer

549 views

### Non-compact Kähler manifolds which admit a positive line bundle

A complex manifold which admits a positive line bundle is automatically Kähler. Furthermore, if the manifold is compact, then it is projective by the Kodaira Embedding Theorem. In particular, not ...

**1**

vote

**0**answers

112 views

### pairing theta functions for different complex structures

I apologize for my previous attempt to ask this, which was very badly written.
Let us start with $\mathbb{C}\times\mathbb{C}$. To form an Hermitian line bundle over a complex torus with complex ...

**1**

vote

**0**answers

317 views

### Theta functions and Fourier transforms

Let $T_\tau$ be the 2-dimensional torus, with the complex structure induced by the lattice generated by $1$ and $\tau$. Then for a line bundle $L_k$ over $T$ with level $k$, there is an orthonormal ...

**7**

votes

**4**answers

1k views

### Cohomology of line bundles

For sure answers to my questions are well known - but I never saw them anywhere.
Let $X$ be a smooth projective (or just proper) variety over an algebraically closed field $k$. Let $A_i$ be the ...